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Viscosity Radius of Polymers in Dilute Solutions: Universal Behaviorfrom DNA Rheology and Brownian Dynamics SimulationsSharadwata Pan,†,‡,§ Deepak Ahirwal,‡ Duc At Nguyen,§ T. Sridhar,†,§ P. Sunthar,†,‡

and J. Ravi Prakash*,†,§

†IITB-Monash Research Academy and ‡Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai,Mumbai 400076, India§Department of Chemical Engineering, Monash University, Melbourne, VIC 3800, Australia

*S Supporting Information

ABSTRACT: The swelling of the viscosity radius, αη, and the universalviscosity ratio, UηR, have been determined experimentally for linear DNAmolecules in dilute solutions with excess salt, and numerically by Browniandynamics simulations, as a function of the solvent quality. In the latterinstance, asymptotic parameter free predictions have been obtained byextrapolating simulation data for finite chains to the long chain limit.Experiments and simulations show a universal crossover for αη and UηR from θto good solvents in line with earlier observations on synthetic polymer−solvent systems. The significant difference between the swelling of thedynamic viscosity radius from the observed swelling of the static radius ofgyration is shown to arise from the presence of hydrodynamic interactions inthe nondraining limit. Simulated values of αη and UηR are in good agreementwith experimental measurements in synthetic polymer solutions reportedpreviously and with the measurements in linear DNA solutions reported here.

1. INTRODUCTIONLarge scale static and dynamic properties of dilute polymersolutions scale as power laws with molecular weight M in thelimits of θ and very good solvents.1,2 In the intermediate regimebetween these two limits, their behavior can be described interms of crossover functions of a single scaling variable, the so-called solvent quality parameter, z = (3/4)K(λL)z, where K is afunction of the chain stiffness parameter (λ−1) and contourlength (L), and the parameter z = k(1 − Tθ/T)√M, combinesthe dependence on temperature T and molecular weight.3−5

The constant k is chemistry dependent, and Tθ is the θ-temperature. In the random coil limit λL → ∞, where polymerchains are completely flexible, z = z. Examples of such crossoverfunctions include the swelling functions, αg = Rg/Rg

θ (which is aratio of the radius of gyration at any temperature T to theradius of gyration at the θ-temperature), αH = RH/RH

θ (whereRH is the hydrodynamic radius), and αη = Rη/Rη

θ = ([η]/[η]θ)

1/3, where Rη is the viscosity radius, defined by theexpression

ηπ

≡η

⎛⎝⎜

⎞⎠⎟R

MN

3[ ]10 A

1/3

(1)

with NA being the Avogadro’s constant and [η] the zero shearrate intrinsic viscosity. Several experimental studies6−9 haveshown that swelling data for many different polymer−solventsystems can be collapsed onto master plots, independent ofchemical details, when represented in terms of the parameter z.

Notably, however, the universal curve for αg (which is a ratio ofstatic properties) is significantly different from the universalcurves for αH and αη, which are ratios of dynamic properties.

6−9

There have been many attempts to understand the origin ofthis difference in crossover behavior and to predict analyticallyand numerically the observed universal curves4,10−18 (see ref 19for a recent review). In this paper, we re-examine this problemin the context of Brownian dynamics (BD) simulations, whichare a means of obtaining an exact (albeit numerical) solution tothe underlying model for the polymer solution. We also reporton experimental measurements of the viscosity radius of DNAin the presence of excess salt (at two different molecularweights) and examine the universality of the crossover ofproperties derived from the viscosity radius by comparison withprevious measurements for synthetic polymer solutions.Dilute polymer solution models typically represent polymers

as chains of beads connected together by rods or springs,immersed in a Newtonian solvent. The beads act as centers offrictional resistance to chain motion through the solvent, andthe motions of all the beads are coupled together throughhydrodynamic interactions which represent the solventmediated propagation of momentum between the beads.Bead overlap is prevented either by excluded volumeinteractions between the beads, acting pairwise through a

Received: May 10, 2014Revised: October 9, 2014Published: October 20, 2014

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© 2014 American Chemical Society 7548 dx.doi.org/10.1021/ma500960f | Macromolecules 2014, 47, 7548−7560

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repulsive potential, or through restriction of chain config-urations to those that are self-avoiding. Within such aframework, analytical theories such as the renormalizationgroup theory,5 and two-parameter theories3 have successfullypredicted static properties of dilute solutions of flexiblepolymers. For instance, both renormalization group and two-parameter theories provide explicit expressions for αg as afunction of z. A well-known example of the latter is the Domb−Barrett equation.19,20

Both the hydrodynamic and viscosity radii are dynamicproperties, and consequently, hydrodynamic interactions play acrucial role in determining the swelling functions αH and αη.Barrett21 used two-parameter theory with preaveraged hydro-dynamic interactions to develop explicit expressions for αH andαη as functions of z. The Barrett equation for αη has proven tobe an extremely accurate means of predicting the swelling of Rη

for a number of different polymer−solvent systems.4,8,19 On theother hand, the Barrett equation for αH considerably over-predicts the extent of swelling of the hydrodynamic radiuswhen compared to experimental measurements in the crossoverregime.4,7,8 Zimm22

first recognized that the neglect offluctuating hydrodynamic interactions in models with pre-averaged hydrodynamic interactions could be a significantfactor responsible for the poor prediction of universalproperties. Yamakawa and co-workers4,15,16 subsequentlydeveloped an approximate analytical model to account for thepresence of fluctuating hydrodynamic interactions within theframework of quasi-two-parameter theory, which is amodification of two-parameter theory that accounts for chainstiffness by introducing the parameter z in place of z as theuniversal scaling variable. They suggest that αH = αH

(0)hH and αη

= αη(0)hη, where αH

(0) and αη(0) are the swelling functions

predicted in the absence of fluctuations and hH and hη arecorrections that account for their presence. Yamakawa andYoshizaki15 have proposed an expression for hH as a function ofz, while currently there is no analytical expression for hη. Theinclusion of fluctuations in hydrodynamic interactions in thismanner leads to a reduction in the values of αH predicted by theBarrett equation; however, they are still too high relative toexperimental values in the entire crossover regime.4,7,8

An alternative explanation17,19 that has been offered for thedifference in the universal crossover functions for αH and αη

from αg is that hydrodynamic interactions are not fullydeveloped in the crossover regime; i.e., rather than being inthe nondraining limit where polymer coils behave as rigidspheres, there is a partial draining of the solvent throughpolymer coils, which are swollen because of excluded volumeinteractions. This approach, however, also does not result in animproved prediction of the universal crossover function forαH.

18

More recently, Sunthar and Prakash18 have shown for flexiblepolymer chains, by carrying out exact BD simulations of bead−spring chains, that the difference between αg and αH is in factdue to the presence of fluctuating hydrodynamic interactions inthe nondraining limit. By accounting for fluctuating hydro-dynamic and excluded volume interactions in the asymptoticlong chain limit, Prakash and co-workers have been able toobtain quantitatively accurate, parameter-free predictions of αgand αH as functions of z.18,23

The agreement of the Barrett equation21 for αη withexperimental observations has been taken to imply that, incontrast to αH, fluctuations in hydrodynamic interactions arenot important in determining the swelling of the viscosity

radius.4,16 However, this cannot be conclusively establishedwithout an exact estimate of the magnitude of fluctuations inthe entire crossover regime. For instance, the agreement couldarise fortuitously from a cancellation of errors due to theassumption of preaveraged hydrodynamic interactions and theoccurrence of partial draining. The use of BD simulationsprovides an opportunity to account exactly for the presence offluctuating hydrodynamic interactions and, consequently, toexamine its role in determining the observed difference in thecrossover of αη and αg, as has been done previously in the caseof αH by Sunthar and Prakash.18

Properties of dilute polymer solutions are often measured inorder to obtain structural information about the dissolvedmacromolecules. By comparing experimental data withpredictions of solution models with different macromolecularstructures, such as flexible, wormlike, ellipsoidal, cylindrical,etc., information on the shape, size, and flexibility ofmacromolecules can be obtained. Rather than using the valuesof properties themselves, it has been found more convenient toconstruct dimensionless ratios of properties, since such ratiostend to depend only on the shape of the macromolecule andnot on its absolute size. A well-known example of such a ratio,based on the intrinsic viscosity and radius of gyration, is theFlory−Fox constant,2 Φ = [η]M/63/2Rg

3. An alternativeapproach proposed by Garcia de la Torre and co-workers isto use equivalent radii instead of properties themselves toconstruct dimensionless ratios.24,25 An equivalent radius isdefined as the radius of a sphere, a dilute suspension of whichwould have the same value of the property as the solution itself.For instance, Rη defined by eq 1 and GI = (5/3)1/2(Rg/Rη) areexamples of an equivalent radius and a nondimensional ratio ofequivalent radii, respectively. Garcia de la Torre and co-workershave shown that the use of such ratios is a more efficient andless error-prone way of extracting structural information.24,25

We use the viscosity ratio, UηR, which is usually defined in thecontext of BD simulations,26,27 as a universal function thatcharacterizes polymer solutions. It is trivially related to both Φand GI

π≡ = Φ =η

η −⎜ ⎟⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝

⎞⎠U

R

R N52

6(4 /3)

52

53

(GI)Rg

33/2

A

3/23

(2)

Kroger et al.27 have tabulated experimentally measured valuesof UηR and the predictions of various approximate theories andsimulations (under both θ solvent and good solventconditions). For θ solvents, experimental measurements28

indicate that UηRθ = 1.49 ± 0.06, which corresponds to the

well-known value of the Flory−Fox constant for flexiblepolymers in θ solvents, Φ0 = 2.56 × 1023. Garcia de la Torreand co-workers25,29−31 have used the Monte Carlo rigid bodymethod, accompanied by extrapolation of finite chain data tothe long chain limit, to predict Φ0 = 2.53 × 1023 in θ solvents(which equates to27 UηR

θ ≈ 1.47 ± 0.15), while in the limit ofvery good solvents (z → ∞) they predict Φ = 1.9 × 1023 (i.e.,UηR

∞ ≈ 1.11 ± 0.10). By carrying out nonequilibrium BDsimulations at finite shear rates, and extrapolating the finiteshear rate data to the limit of zero shear rate, Kroger et al.27

predict UηRθ ≈ 1.55 ± 0.04. Jamieson and Simha19 observe that

even though a number of experimental measurements of theFlory−Fox constant under good solvent conditions have beenreported in the literature, the behavior of Φ with varyingsolvent conditions and molecular weight appears not to beunderstood with any great certainty.

Macromolecules Article

dx.doi.org/10.1021/ma500960f | Macromolecules 2014, 47, 7548−75607549

An analytical expression for the crossover behavior of theratio UηR/UηR

θ (which is also equal to the ratio of the Flory−Foxconstants in good and θ-solvents) can be determined bysubstituting the Domb−Barrett equation20 for αg and theBarrett equation21 for αη on the right-hand side of the followingexpression (which follows from the definitions of the variousquantities involved):

αα

=η

ηθ

η⎛⎝⎜⎜

⎞⎠⎟⎟

U

UR

R g

3

(3)

Not surprisingly, given the accuracy of the Domb−Barrett andBarrett equations, experimental data on the crossover of thisratio are well captured by quasi-two-parameter theory.8,19

However, as in the case of the expansion factor αη, so far noexact Brownian dynamics simulations have been carried out todetermine the crossover behavior of UηR (a knowledge of whichwould also provide the ratio UηR/UηR

θ ).Reported observations of αη and Φ have largely been on

synthetic polymer−solvent systems.6,8,9 Recently, Pan et al.32

have shown that the crossover swelling of the hydrodynamicradius of linear DNA molecules in dilute solutions with excesssalt can be collapsed onto earlier observations of the swelling ofthe hydrodynamic radius of synthetic polymers. This result wasestablished by (i) showing with the help of static light scatteringthat the θ-temperature of a commonly used excess salt solutionof linear DNA molecules is Tθ ≈ 15 °C and (ii) estimating thehydrodynamic radius and the solvent quality at any temperatureand molecular weight by dynamic light scattering measure-ments. These developments make it now possible to examinethe crossover behavior of any static or dynamic property oflinear DNA solutions in the presence of excess salt.The aim of this paper is twofold: (i) To carry out systematic

measurements of the intrinsic viscosity of two differentmolecular weight samples of linear double-stranded DNA at arange of temperatures in the presence of excess salt andexamine the crossover scaling of the swelling of the viscosityradius, αη, and the viscosity ratio, UηR. Comparison with earlierobservations of the behavior of synthetic polymers not onlyenables the establishment of the universal scaling of DNAsolutions but also serves as an independent verification of theearlier estimate of the θ-temperature and solvent quality by Panet al.32 (ii) To carry out detailed BD simulations of bead−spring chains to estimate αη and UηR as functions of z forflexible polymers. This has previously been difficult because ofthe large error associated with simulations of viscosity at lowshear rates. By using a Green−Kubo formulation and a variancereduction scheme, coupled with systematic extrapolation offinite chain data to the long chain limit to circumvent theproblem of poor statistics, we show for the first time that byincluding fluctuating excluded volume and hydrodynamicinteractions, a quantitatively accurate prediction of thecrossover scaling of αη and UηR can be obtained, free fromthe choice of arbitrary model parameters. Further, thedifference between the crossover scaling of αg and αη isshown to arise undoubtedly from the influence of hydro-dynamic interactions in the nondraining limit, and the relativeunimportance of fluctuations in hydrodynamic interactions isconfirmed.The plan of the paper is as follows. In section 2, we describe

the experimental protocol for preparing the DNA samples andfor carrying out the viscosity measurements. We also discussthe governing equations for the BD simulations, the variance

reduction scheme adopted here, and the calculation of theviscosity using a Green−Kubo expression. In section 3.1, wedescribe the measurement of the intrinsic viscosity of the DNAsolutions and tabulate values of intrinsic viscosity and theHuggins coefficient across a range of temperatures. In theremaining subsections of section 3, we discuss the prediction ofαη and UηR by BD simulations and compare simulationpredictions with prior and current experimental measurements.Finally, in section 4, we summarize the principal conclusions ofthe present work.

2. MATERIALS AND METHODS2.1. DNA Samples and Shear Rheometry. Viscosities have been

measured for two different double-stranded DNA molecular weightsamples: (i) T4 bacteriophage linear genomic DNA [size 165.6kilobasepairs (kbp)] and (ii) 25 kbp DNA. While the former wereobtained from Nippon Gene, Japan (#314-03973), the latter wereextracted, linearized, and purified from Escherichia coli (E. coli) stabcultures procured from Smith’s laboratory at UCSD. Smith’s grouphave genetically engineered special double-stranded DNA fragments inthe range of 3−300 kbp and incorporated them inside commonly usedE. coli bacterial strains. These strains can be cultured to producesufficient replicas of its DNA, which can be cut precisely at desiredlocations to extract the special fragments.33 The protocol for preparingthe 25 kbp samples obtained in this manner has been described indetail in Pan et al.32 Typical properties of the DNA molecules used inthis work, such as the molecular weight, contour length, number ofKuhn steps, etc., are tabulated in Table S-1 (Supporting Information).Additionally, details regarding the solvent, estimation of DNAconcentration, etc., are presented in the Supporting Information.

A Contraves Low Shear 30 rheometer has been used to obtain allthe shear viscosity measurements reported in the present work becauseof two main advantages: it has a zero shear rate viscosity sensitivityeven at a shear rate of 0.017 s−1, and thus can measure very lowviscosities, and has a very small sample requirement (minimum 800μL).34 Both of these are ideal for measuring viscosities of biologicalsamples such as DNA solutions. The steady state shear viscosities ηwere measured at low concentrations (c < c*) and across atemperature range of 15−35 °C. The overlap concentrations (c*), atdifferent temperatures, were estimated from the known values of thesolvent quality parameter z, as described in Pan et al.32 The zero shearrate viscosity was determined from measurements of viscosity atdifferent finite shear rates and extrapolation to zero shear rate. Detailsare given in the Supporting Information. Values obtained this way forthe two molecular weights, across the range of concentrations andtemperatures, are displayed in Table S-2 (Supporting Information).

2.2. Brownian Dynamics Simulations. The dilute polymersolution is modeled as an ensemble of noninteracting bead−springchains, immersed in a Newtonian solvent. Each chain has N beads ofradius a, connected together by Hookean springs with spring constantH. The beads act as centers of frictional resistance, with a Stokesfriction coefficient, ζ = 6πηsa (where ηs is the solvent viscosity), andbead overlap is prevented through a pairwise repulsive narrowGaussian excluded volume potential (which is a regularization of adelta function potential). Hydrodynamic interactions between thebeads are modeled with the Rotne−Prager−Yamakawa (RPY)regularization of the Oseen function. Within this framework, thetime evolution of the positions of the N beads, r1(t), r2(t), ..., rN(t), isgoverned by stochastic differential equations, which can be integratednumerically (exactly) with the help of Brownian dynamics simulations.Details of the stochastic differential equations, the precise forms of theexcluded volume potential and hydrodynamic interaction tensor, andkey aspects of the integration algorithm are given in the SupportingInformation. It is sufficient to note here that by adopting the lengthscale lH = (kBT/H)

1/2 and time scale λH = ζ/4H for the purpose ofnondimensionalization (where kB is Boltzmann’s constant), it can beshown that there are three parameters that control the dynamics offinite bead−spring chains at equilibrium, namely, the number of beads



N, the strength of excluded volume interactions z*, and thehydrodynamic interaction parameter, h* = a(H/(πkBT))

1/2.Analytical theories have shown that the true strength of hydro-

dynamic interactions is determined by the draining parameter,35,36 h =h*√N, while for flexible polymers, the strength of excluded volumeinteractions is determined by the excluded volume parameter,5,37 z =z*√N. Note that the experimentally measured solvent qualityparameter defined previously for flexible chains can be mapped ontotheoretical values of z by a suitable choice of the constant k.23

Universal predictions, independent of details of the coarse-grainedmodel used to represent a polymer, are obtained in the limit of longchains, since the self-similar character of real polymer molecules iscaptured in this limit. It is common to obtain predictions in the longchain limit by accumulating data for finite chain lengths andextrapolating to the limit N → ∞. This procedure has been usedsuccessfully to calculate universal properties of dilute polymersolutions predicted by a variety of approaches to treating hydro-dynamic and excluded volume interactions, including Monte Carlosimulations,22,29−31 approximate closure approximations,38−41 andexact Brownian dynamics simulations.18,23,27,42−44

The nondraining limit corresponds to h → ∞. As a result,simulations carried out at constant values of h* naturally lead topredictions in the nondraining limit as N→ ∞. Sunthar and Prakash18

have shown that universal predictions in the nondraining limit, and atany fixed value of the solvent quality parameter z = z*√N, can beobtained by simultaneously keeping h* and z constant, while takingthe limit N → ∞. Since the parameter z* → 0 in this limit, the longchain limit of the model corresponds to the Edwards continuous chainmodel with a delta function excluded volume repulsive potential.45 Asmentioned in section 1, by accounting for fluctuating hydrodynamicand excluded volume interactions in this manner, Sunthar andPrakash18 have obtained a quantitatively accurate parameter-freeprediction of αH as a function of z. Here, we show that this approachcan also be used to successfully predict universal properties related tothe zero shear rate viscosity of dilute polymer solutions.2.3. Universal Properties Derived from the Viscosity Radius.

We focus our attention on two properties that are defined in terms ofthe viscosity radius (eq 1) which have been shown to be universal inthe sense that they are independent of the chemistry of the particularpolymer−solvent system for sufficiently long polymers. The first ofthese is the universal viscosity ratio, UηR (defined in eq 2), and thesecond is the swelling ratio, αη. We discuss the evaluation of theseproperties by Brownian dynamics simulations in turn below.In terms of dimensionless variables, UηR can be shown to be given

by

πη

= **

*ηU hR

98R

p,0

g3

(4)

where Rg* is the dimensionless radius of gyration and ηp,0* = ηp,0/(npλHkBT) is the dimensionless zero-shear rate viscosity. Here, np is thenumber of chains per unit volume, and ηp,0 = η0 − ηs is the polymercontribution to the zero shear rate solution viscosity. Kroger et al.27

have estimated ηp,0* by carrying out nonequilibrium BD simulations atfinite shear rates and extrapolating the data to the limit of zero shearrate. Here, we use an alternative method based on a Green−Kuborelation46 which gives the viscosity as an integral of the equilibrium-averaged stress−stress autocorrelation function

∫η* = ⟨ ⟩∞

t C tr r rd ( , , ..., , )Np,0 0S 1 2 eq (5)

where

=C t S t Sr r r( , , ..., , ) ( ) (0)N xy xyS 1 2 (6)

The quantity Sxy is the xy-component of the stress tensor given byKramers expression Sxy = ∑μFμx(rμy − rcy), where rμy is the y-component of rμ, rcy is the y-component of the position vector of thecenter-of-mass of the bead−spring chain, rc = (1/N)∑μrμ, and Fμx isthe x-component of Fμ, the sum of all the nonhydrodynamic forces on

bead μ due to all the other beads. The use of the Green−Kubo methodmitigates the problem of the large error bars associated with estimatingpolymer solution properties at low shear rates. We find that the noisein measured properties can be significantly reduced by evaluating theintegral in eq 5 with the help of equilibrium simulations of a largeensemble of trajectories. Additionally, for some simulations, we haveemployed a variance reduction technique, as explained in section 2.4.

Rather than evaluating the swelling of the viscosity radius directlyfrom its definition αη = Rη/Rη

θ, we found it advantageous to use thefollowing expression (obtained by rearranging eq 3), which gives αη interms of UηR and αg, since the N → ∞ extrapolations of UηR

θ and UηR(at various values of z) are more accurate than the extrapolations forαη

α α=ηη

ηθ

⎛⎝⎜⎜

⎞⎠⎟⎟

U

UR

R

1/3

g(7)

The swelling of the radius of gyration αg for different values of z iscalculated from the expression

α = + + +az bz cz(1 )mg

2 3 /2(8)

with values of the fitting parameters a, b, c, and m, as given in Table 3.This specific form for the fitting function is often used inrenormalization group theory predictions and in lattice simulationsto represent the crossover behavior of swelling ratios for flexiblechains.5 Equation 8 has been shown by Kumar and Prakash23 to be anexcellent fit to the asymptotic predictions of αg by BD simulations inthe absence of hydrodynamic interactions. This corresponds to thepure excluded volume problem, which is adequate for determining αg,since it is a static property unaffected by hydrodynamic interactions.

2.4. Variance Reduced Simulations. The statistical error in theestimation of the equilibrium-averaged stress−stress autocorrelationfunction ⟨CS(t)⟩eq can be significantly reduced if the fluctuations inCS(r1, r2, ..., rN, t) can be made to be small. Among the manyapproaches available for reducing the magnitude of fluctuations instochastic simulations,26 we have adopted a variance reductiontechnique based on the use of control variates,47 as described below.

In general, the fluctuations f CS= CS(r1, r2, ..., rN, t) − ⟨CS(t)⟩eq

cannot be estimated a priori. However, if the fluctuations fCS= CS(r1,

r2, ..., rN, t) − ⟨CS(t)⟩eq can be determined for a stochastic process rvfor which the equilibrium-averaged stress−stress autocorrelation⟨CS(t)⟩eq is known analytically, and fCS

≈ f CS, then the control variate

= − E C t fr r r( , , ..., , )C N CS 1 2S S (9)

can be used to estimate the stress−stress autocorrelation function withreduced statistical error, since ⟨ECS

⟩eq = ⟨CS(t)⟩eq. The extent of thereduction in statistical error depends on the extent to which CS and CSare correlated, as can be seen from the expression for the variance ofECS

⟨ − ⟨ ⟩ ⟩ = ⟨ − ⟨ ⟩ ⟩ + ⟨ − ⟨ ⟩ ⟩

− ⟨ ⟩ − ⟨ ⟩ ⟨ ⟩

E E C C C C

C C C C

[ ] [ ] [ ]

2[ ]

C C eq2

eq S S eq2

eq S S eq2

eq

S S eq S eq S eq

S S

(10)

We use the stochastic process rv, governed by the stochastic differentialequation

∑ ∑ = +μν

μνν

μν νH t Sr F Wd14

d12

dv(11)

as a trajectory-wise approximation to rν. Here Wν is a Wiener process,and the N × N matrix Hμν is the equilibrium average of the diffusiontensor Dμν (see Supporting Information), given by

δ δ= + − μν μν μν μνH H(1 ) (12)

The expression for the matrix Hμν is discussed shortly below. Thematrix Sμν satisfies the expression



∑ μ ν= ≠α

μα να μνS S H , for(13)

Note that Hμμ = Sμμ = 1. The equilibrium average of Dμν is carried outwith the equilibrium distribution function in the absence of excludedvolume interactions, since an analytical solution for the distributionfunction is only known under θ-solvent conditions. The advantage ofusing eq 11 for the purpose of variance reduction comes from the factthat Fixman has previously calculated Hμν and ⟨CS(t)⟩eq analytically forthe RPY tensor.46,48 By simulating eq 11 simultaneously with thestochastic differential equation for rν (see Supporting Information),with the same Weiner process Wν, the fluctuations fCS

can be estimated

and consequently the mean value of the control variate, ⟨ECS⟩eq. For

the sake of completeness, we reproduce Fixman’s expressions for Hμν

and ⟨CS(t)⟩eq with the nondimensionalization scheme and notationused here, in the Supporting Information.The efficacy of the variance reduction procedure is demonstrated in

Figure 1, where the various autocorrelation functions obtained from

the simulation of a bead−spring chain under θ-conditions, with N = 18and h* = 0.25, are displayed. The positive correlation between the twofunctions CS and CS and the reduction in the variance in ECS

can be

clearly observed.Variance reduction was used here only for simulations with z = 0 (θ-

solvent), z = 0.01, and z = 0.1. For higher z, the correlation betweenthe two stochastic processes was lost, and there was no benefit in usingECS

in place of CS. This is not unexpected since the equilibrium

averaging of the diffusion tensor is carried out with the equilibriumdistribution function in the absence of excluded volume interactions.The stress−stress autocorrelation function must be integrated to

obtain the intrinsic viscosity, as can be seen from eq 5, where, whenappropriate, we use the control variate ECS

(t) instead of CS(t). In spite

of the reduced variance, the numerical integration of this function issubject to errors. Consequently, we use a nonlinear least-squares fit ofthe autocorrelation function instead and evaluate the integral of thefitting function. Details are given in the Supporting Information.

3. RESULTS AND DISCUSSION3.1. Intrinsic Viscosity of DNA Solutions. The intrinsic

viscosity of a polymer solution is typically obtained from a virialexpansion of the dilute solution viscosity as a function ofconcentration. Two commonly used forms of the virialexpansion are the Huggins equation

ηη

ηη η η≡ = + + ′ +c k c k c[ ] ([ ] ) ([ ] ) ...sp

p,0

sH

2H

3

(14)

and Kraemer’s equation

ηη

η η η= − + ′ +c k c k cln [ ] ([ ] ) ([ ] ) ...0

sK

2K

3

(15)

where ηsp is the specific viscosity and the coefficient kH in thequadratic term in Huggins’ equation (eq 14) is the Hugginsconstant, and is analogous to the second virial coefficient forviscosity,2 while kK is the equivalent coefficient in Kraemer’sequation. The parameters kH′ and kK′ are coefficients of the cubicterms in the Huggins’ and Kraemer’s equations, respectively.Substituting the Huggins expansion in terms of η0 from eq 14

into the left-hand side of Kraemer’s equation (eq 15) andcomparing terms of similar order leads to

= − ′ = ′ − +k k k k k12

and13K H K H H (16)

Typically, dilute solution viscosities are measured at low valuesof concentration, where the contribution of the cubic term inthe Huggins equation is negligible. As a result, by plotting ηsp/cversus concentration, the intrinsic viscosity can be obtainedfrom the intercept on the y-axis of a straight line fitted to thedata, while kH can be determined from the slope of the line,since

ηη η= +

ck c[ ] [ ]sp

H2

(17)

As pointed out by Pamies et al.,50 even though k′H([η] c)3 ≈ 0,the contribution of the cubic term in Kraemer’s equation neednot be zero (unless, kH ≈ 1/3, see eq 16). At sufficiently lowconcentrations, however, Kraemer’s equation (eq 15) suggeststhat [ln(η0/ηs)]/c will be linear in concentration

ηη

η η= −c

k c1

ln [ ] [ ]0

sK

2

(18)

As a result, the intrinsic viscosity can be obtained from theintercept of a line fitted to measurements of [ln(η0/ηs)]/cversus c (in a so-called Fuoss−Mead plot51), while kK can bedetermined from the slope of the line.Since the leading order term in the expansions for both ηsp

and ln(η0/ηs) is [η]c, Solomon and Ciuta 52 suggested that thevirial expansion of the difference ηsp − ln(η0/ηs) would have aweaker dependence on concentration

ηηη

η η− = + ′ +k c k cln ([ ] ) ([ ] ) ...sp0

sSC

2SC

3

(19)

with

= ′ = −k k k12

and13SC SC H (20)

As a result, by defining the quantity

Figure 1. Reduction in the variance of the stress autocorrelationfunction. The two autocorrelation functions, CS (red curve) calculatedwith fluctuating hydrodynamic interactions and CS (blue curve)calculated with preaveraged hydrodynamic interactions, can be seenvisually to be positively correlated. The control variate ECS

(green

curve) clearly has significantly lower fluctuations. The analyticalfunction ⟨CS(t)⟩eq (black curve) is given by Fixman’s expression46 (seeSupporting Information). The range of the axes have been chosen tomagnify the noise at small values of CS. In this simulation λ1 = 38.2 isthe longest relaxation time, estimated from Thurston’s correlation49

for N = 18, and h* = 0.25. The averages have been obtained overroughly 57 000 independent trajectories.




η η η η= −c

[ ]1

2( ln( / ))c sp 0 s (21)it follows that

η η η= + ′ +k c[ ] [ ] [ ] ...c SC2

(22)

Figure 2. Determination of [η] for 25 kbp and T4 DNA. The left and right column of figures represent 25 kbp and T4 DNA, respectively, atdifferent temperatures (indicated within the figures). The solid, dashed, and dotted lines are least-squares linear fits to the data points extrapolated tozero concentration in accordance with the Huggins, Kraemer, and Solomon−Ciuta equations, respectively. In each figure, the mean value of [η]obtained by extrapolating data for [ln(η0/ηs)]/c (open diamonds), ηp,0/cηs (filled squares) and [η]c (half-filled triangles) to zero concentration, isrepresented by a filled circle (the common intercept on the y-axis). Note that the quantities on the y-axis are in units of mL/mg, the same as [η].

Table 1. Intrinsic Viscosities [η] (in mL/mg) for 25 kbp and T4 DNA at Various Temperatures (T), As Obtained from DifferentExtrapolation Methods: Huggins ([η]H), Kraemer ([η]K), and Solomon−Ciuta ([η]SC)a

25 kbp T4 DNA

T (°C) [η]H [η]K [η]SC [η]mean αη [η]H [η]K [η]SC [η]mean αη

15 7.6 ± 0.1 7.4 ± 0.1 7.5 ± 0.1 7.5 ± 0.4 1 ± 0.03 28.5 ± 1.4 28.9 ± 1.3 28.8 ± 1.3 28.7 ± 3.1 1 ± 0.0518 8.3 ± 0.5 8.3 ± 0.4 8.4 ± 0.4 8.3 ± 0.9 1.03 ± 0.0420 44.2 ± 0.7 44.3 ± 0.6 44.3 ± 0.6 44.3 ± 1.5 1.15 ± 0.0421 9.4 ± 0.3 9.3 ± 0.2 9.3 ± 0.2 9.3 ± 0.5 1.07 ± 0.0325 9.9 ± 0.1 9.7 ± 0.1 9.8 ± 0.1 9.8 ± 0.3 1.09 ± 0.02 57.1 ± 2.4 56.6 ± 1.7 57 ± 2 56.9 ± 4.6 1.26 ± 0.0630 13.2 ± 0.2 12.4 ± 0.1 12.7 ± 0.1 12.8 ± 1.1 1.19 ± 0.04 69.7 ± 1.5 68.7 ± 0.8 69.3 ± 1.1 69.2 ± 2.2 1.34 ± 0.0535 14.2 ± 0.4 13.5 ± 0.1 13.8 ± 0.2 13.8 ± 1.2 1.22 ± 0.04 77.5 ± 5.3 76.8 ± 3.7 77.5 ± 4.1 77.3 ± 9.8 1.39 ± 0.08

aThe mean of the [η] values from these extrapolations is also indicated at each temperature. The swelling ratio αη is also listed for each DNA at eachtemperature and has been calculated based on the [η]mean values. Note that Tθ = 15 °C.




As discussed in some detail by Pamies et al.,50 under the specialcircumstances when kSC′ [η]2c ≈ 0 or kH ≈ 1/3 (see eq 20), theintrinsic viscosity can be determined from the Solomon−Ciuta equation (eq 22) by measuring the viscosity at a singleconcentration, without the necessity of an extrapolationprocedure. The departure of [η]c from a constant value when[η]c is plotted as a function of c can be seen as indicating thedeparture of kH from a value of 1/3.Plots of the relevant variables in the linear versions of the

Huggins equation (eq 17), the Kraemer equation (eq 18), andthe Solomon−Ciuta equation (eq 22), as a function ofconcentration, can now be interpreted in the light of thediscussion above. Figure 2 displays plots of ηsp/c, [ln(η0/ηs)]/c,and [η]c, obtained using results of the zero shear rate solutionviscosity measurements, as a function of concentration. Valuesof [η] obtained by extrapolating linear fits to the finiteconcentration data to the limit of zero concentration are listedin Table 1, where the subscript on [η] indicates the equationused to obtain the value. The mean values of [η] obtained fromthe three methods are also indicated in the table. It is clear thatthe three extrapolation methods give values that are fairly closeto each other.Recently, Rushing and Hester53 have shown that, in line with

a relationship proposed originally by Stockmayer and Fixman,54

the ratio ([η]/M) for a number of different polymer−solvent

systems scales linearly with inverse temperature, with a slopethat is independent of molecular weight. Figure 3 indicates thatthe mean value of [η]/M, for both the DNA samples, scales

linearly with inverse temperature as T increases from Tθ togood solvent conditions, with a slope that is common for boththe DNA, in agreement with the observations of Rushing andHester53 for synthetic polymer solutions.As discussed earlier, the values of kH can be obtained from

the slopes of the lines in Figure 2. While it is obtained directlyfrom the slope of the line through the Huggins data, Kraemer’sdata give kH from kK (see eq 16), and the Solomon−Ciuta datagive kH from kSC′ (see eq 20). The values of kH obtained fromthese different methods are listed in Table 2. We first discussthe data for T4 DNA, which appear to be more in line withprevious observations on synthetic polymer solutions.Pamies et al.50 have recently tabulated values of kH for several

systems by collating data reported previously in the literature(see Table 1 in ref 50). For flexible polymers, kH is observed tolie in the range 0.4−0.7 for θ-solvents and in the range 0.2−0.4for good solvents. Clearly, values of kH reported for T4 DNA inTable 2 lie in the expected ranges for θ and good solvents, withthe θ-solvent value greater than that for good solvents. Thethree different means of estimating kH also give valuesreasonably close to each other. Since kH ≈ 1/3, we expectfrom the Solomon−Ciuta equation (eq 22) that the slope ofthe line through measured values of [η]c as a function ofconcentration should be close to zero. This is indeed the case,as can be seen from parts b, d, and f for T4 DNA in Figure 2.When the term of order ([η]c)3 is negligible, we expect a plot

of ηsp versus c[η] to depend quadratically on c[η] for increasingvalues of c[η] (see eq 14). The departure from linearity can beobserved for the T4 DNA data in Figure 4a for c[η] ≳ 0.3(filled symbols). The importance of the quadratic term can beseen more clearly by plotting ηsp/(c[η]) versus (c[η]), as shownin Figure 4b, since

η

ηη= +

ck c

[ ]1 [ ]sp

H(23)

The data for T4 DNA is scattered around a line with slope = 1/3, as expected from the values of kH listed for T4 DNA in Table2.Values of kH extracted from the dilute solution viscosity data

for 25 kbp DNA using the Huggins method have a greaterdegree of uncertainty associated with them compared to thosefor T4 DNA (see first column in Table 2). Even though thevalues obtained from the Kraemer and Solomon−Ciuta equations lie closer to the expected range of values for goodsolvents, the θ-solvent values are smaller than the good solventvalues. Figure 4a indicates that the dependence of ηsp on c[η]Hfor 25 kbp DNA appears to be linear in the entire range ofvalues of c[η]H observed here (empty symbols), which suggeststhat it would be harder to extract the values of kH with

Figure 3. Temperature dependence of [η]/M for 25 kbp DNA and T4DNA. The line through the T4 data is a least-squares linear fit, whilethe line through the 25 kbp data, which is more scattered, is drawnwith the same slope to guide the eye.

Table 2. kH As Obtained from the Huggins, Kraemer, and Solomon−Ciuta Equations for 25 kbp and T4 DNA at DifferentTemperatures

kH (Huggins) kH (from Kraemer, see eq 16) kH (from Solomon−Ciuta, see eq 20)

T (°C) 25 kbp T4 DNA 25 kbp T4 DNA 25 kbp T4 DNA

15 (Tθ) 0.06 ± 0.04 0.82 ± 0.22 0.19 ± 0.02 0.64 ± 0.18 0.14 ± 0.3 0.68 ± 0.1918 0.24 ± 0.13 0.28 ± 0.09 0.25 ± 0.120 0.35 ± 0.05 0.35 ± 0.03 0.33 ± 0.0421 0.24 ± 0.05 0.3 ± 0.03 0.26 ± 0.0425 0.16 ± 0.01 0.24 ± 0.09 0.27 ± 0.01 0.29 ± 0.06 0.21 ± 0.01 0.26 ± 0.0730 0.01 ± 0.02 0.23 ± 0.04 0.22 ± 0.01 0.3 ± 0.02 0.14 ± 0.01 0.26 ± 0.0335 0.08 ± 0.03 0.32 ± 0.12 0.26 ± 0.01 0.35 ± 0.08 0.18 ± 0.02 0.31 ± 0.08




confidence using the Huggins method. This is also clearlyreflected in Figure 4b, where the data indicate that the value ofthe Huggins constant is highly scattered and in most casessmaller than 1/3. More extensive measurements at a largerrange of concentrations would be required to obtain kH withgreater accuracy for 25 kbp DNA.The intrinsic viscosity data obtained at various temperatures

can be used to calculate the viscosity radius of 25 kbp and T4DNA. Of the two properties of interest in the present work,namely, UηR and αη, the latter is directly calculable fromexperimental measurements. Values for the two DNA samplesare reported in Table 1. On the other hand, the directestimation of UηR requires the additional knowledge of Rg.While the prediction of UηR here by simulations is based on thedetermination of both the viscosity and the radius of gyration asa function of solvent quality, we do not have experimentalinformation on Rg for the two DNA samples studied here.However, it is clear from eq 3 that the ratio (UηR/UηR

θ ) can becalculated without a knowledge of Rg, if the dependence of αη

and αg on solvent quality is known.In the context of determining the dependence of αH on

solvent quality for DNA, Pan et al.32 established therelationship between pairs of values of T and M, and z,assuming that DNA is a flexible molecule at the molecularweights that were considered. Here, we take into account thewormlike nature of DNA molecules and show in theSupporting Information that a mapping between T and Mand the parameter z can be constructed, similarly. As a result,since the swelling αη is known for the two DNA samples atvarious values of T (Table 1), we can determine thedependence of αη on z for these two samples. Thedetermination of the dependence of αg on z is discussed below.As mentioned earlier, the quasi-two-parameter theory is an

extension of the two-parameter theory to account for chainstiffness.4 Essentially, the theory assumes that functional formsof universal crossover functions for wormlike chains areidentical to those for flexible chains, with the excluded volumeparameter z replaced by the parameter z. As a consequence, thequasi-two-parameter theory expects the Domb−Barrett andBarrett equations for αg and αη, respectively, to successfullydescribe the swelling of the radius of gyration and the viscosityradius of wormlike chains, when z is replaced by z. This

expectation has been shown to be exceedingly well fulfilled for arange of experimental data for a variety of polymer−solventsystems.8,55 Here, we assume analogously that the functionalform used to fit BD data for the swelling of the radius ofgyration of flexible chains can be used to describe the swellingof wormlike chains by replacing z with z. As a result, thedependence of αg on z can be obtained from eq 8, and theexperimentally measured dependence of (UηR/UηR

θ ) on z can bedetermined from eq 3, using experimentally measured values ofαη and BD simulation results for αg.The procedure outlined above enables a comparison of

experimentally measured values of αη and (UηR/UηRθ ) for DNA,

at identical values of the solvent quality z, with earlierobservations for synthetic polymer solutions and with resultsof Brownian dynamics simulations, as discussed in the followingsections.

3.2. Universal Viscosity Ratio under θ-Conditions. Thezero shear rate viscosity, in the absence of hydrodynamicinteractions, is related to the radius of gyration by theexpression

η* = *NR23p,0 g

2

(24)

which can be derived by developing a retarded motionexpansion for the stress tensor.37 As a result, ηp,0 scales withN as N2, and in the absence of hydrodynamic interactions, theratio UηR

θ is not a universal constant since it scales with N asN1/2 (see eq 4). It becomes a universal constant only whenhydrodynamic interactions are included in the model since thisalters the scaling of ηp,0 with N from N2 to N3/2, as firstdemonstrated by Zimm theory2 and by two-parameter theorieswhich include preaveraged hydrodynamic interactions.21

The framework for getting universal predictions within thecontext of Brownian dynamics simulations that includefluctuating hydrodynamic interactions has been clearlydelineated by Kroger et al.,27 who show that model-independent predictions of several properties can be obtainedby careful extrapolation of data accumulated for finite chains tothe long chain limit. By carrying out nonequilibrium Browniandynamics simulations at finite shear rates, and by extrapolatingthe finite shear rate data to the limit of zero shear rate, theyhave obtained equilibrium predictions of several properties. In

Figure 4. (a) Dependence of the specific viscosity ηsp on the nondimensional concentration c[η]H and (b) dependence of the dimensionless ratioηsp/c[η]H on c[η]H for the two DNA used in this work at different absolute concentrations, each of which is at different temperatures in goodsolvents.




particular, they predict UηRθ ≈ 1.55 ± 0.04. In contrast to their

approach, we have used a Green−Kubo expression (eq 5)coupled with a variance reduction scheme in order to obtainpredictions of the zero shear rate viscosity under θ-solvent

conditions. Results for UηRθ obtained by following this

procedure are displayed in Figure 5, where data at constanth*, at several different chain lengths N, are extrapolated to N→∞, which corresponds to the nondraining limit. The choice of1/√N as the x-axis is made because the leading ordercorrection to the infinite chain length limit value of universalratios has been shown to be (1/√N) in Zimm theory38,56

and in simulations.27 As is well-known,36,38 there is a specialvalue of h* called the fixed point, denoted by hf*, at which theleading order correction to the limiting value changes frombeing of (1/√N) to (1/N), resulting in the asymptoticvalue being attained for smaller values of N. For preaveragedhydrodynamic interactions, it is known that hf* = 0.2424...38,56

It is also known that calculations of universal properties forvalues of h* above and below hf* approach the long chain limitvalue along curves with slopes of opposite sign with increasingvalues of N. The choice of values of h* in the currentsimulations have been motivated by these considerations, inorder to obtain better estimates of long chain limit predictions.As can be seen from Figure 5, values of UηR

θ for h* = 0.2 and h*= 0.25 approach the long chain limit along curves whose slopesare of opposite sign to those for h* = 0.45 and h* = 0.5. Thissuggests that for simulations predictions of UηR

θ with fluctuatinghydrodynamic interactions, hf* > 0.25.Extrapolated values of UηR

θ obtained from the currentsimulations, for each h*, have been averaged along with theerror bars to obtain UηR

θ = 1.49 ± 0.10, which is in closeagreement with the experimental value of 1.49 ± 0.06 reportedby Miyaki et al.28 and with the simulation result of 1.47 ± 0.15predicted by Garcia de la Torre et al.29 using Monte Carlo rigid

body simulations. A comparison between UηRθ predictions from

current simulations with results of the simulations of Kroger etal.27 is also shown in Figure 5. It is clear that the scatter in thevalues obtained from an extrapolation of finite shear rate data issignificantly more than that obtained using the method adoptedin the present work.

3.3. Solvent Quality Crossover of UηR. The presenttechnique of extrapolating finite chain data to the long chainlimit, while simultaneously keeping h* and z constant, leads toasymptotic predictions of the crossover behavior of flexiblechains in the nondraining limit. Figure 6 displays the results ofadopting this procedure to predict the crossover behavior ofUηR. At each value of z, data are accumulated at fixed values ofh* for several values of chain length N. The mean of theextrapolated values of UηR in the long chain limit, for thedifferent h*, is considered to be the universal value of UηR atthat value of z. Legends in Figure 6a−d indicate the asymptoticvalues of the universal ratio obtained at the respective values ofz.Figure 7 displays the dependence on z of the asymptotic

values of UηR obtained in this manner. Starting at UηRθ = 1.49 ±

0.1, at z = 0, the universal ratio appears to decrease rapidly withincreasing values of z, leveling off to an excluded volume limitvalue of UηR

∞ = 1.1 ± 0.1, for z ≳ 5. Experimental observationsof the dependence of the Flory−Fox constant on solventquality for a number of different polymer−solvent systems havebeen summarized in the recent review by Jamieson andSimha.19 The general consensus appears to be that Φ decreasesrapidly with increasing solvent quality and with increasingmolecular weight in good solvents. The behavior displayed inFigure 7 is in agreement with the qualitative trend expectedfrom experimental observations.19 Further, the value UηR

∞ = 1.1± 0.1 is in excellent agreement with the earlier prediction of1.11 ± 0.10 by Garcia Bernal et al.31 in the good solvent limit.As will be discussed in greater detail in section 3.4, the

dependence of the swelling αη on the solvent quality z,predicted by Brownian dynamics simulations, can berepresented by a functional form identical to that for αg in eq8, with values of the parameters a, b, and c as given in Table 3.The value of the exponent m, however, is the same in theexpressions for both the crossover functions αη and αg, since (ascan be seen from eq 3) this must be true in order for UηR tolevel off to a constant value for large values of z, as observed inthe BD simulations displayed in Figure 7. Using the functionalforms for αη and αg, and eq 3, it follows that

=+ + ++ + +η η

θ η η η⎛⎝⎜⎜

⎞⎠⎟⎟U U

a z b z c z

a z b z c z

1

1R R

m2 3

g g2

g3

3 /2

(25)

where the suffixes on the parameters a, b, and c indicate therelevant crossover function. The red curve in Figure 7 is a fit tothe BD simulation data using eq 25, along with UηR

θ = 1.49, andthe appropriate values for the fitting parameters listed in Table3. Clearly the fit is very good, as can be expected from theexcellence of the fits for the crossover functions for αη and αg.Tominaga et al.8 have reported experimental measurements

of the dependence of αη on z and have also plotted log αη3

versus log αg3 for a number of different wormlike polymer−

solvent systems. Consequently, using eq 3, the dependence of(UηR/UηR

θ ) on z can be determined for all the experimentalsystems studied in ref 8. As discussed earlier in section 3.1, thisratio can also be determined, as a function of z, for the 25 kbpand T4 DNA samples studied here. Figure 8 displays the data

Figure 5. Universal viscosity ratio for a θ-solvent (UηRθ ). The filled

symbols are the results of current BD simulations determined usingthe Green−Kubo expression for the zero shear rate viscosity: (■) h* =0.2, (▲) h* = 0.25, (◀) h* = 0.45, and (▶) h* = 0.5. The emptysymbols are the results of nonequilibrium simulations at finite shearrate reproduced from Kroger et al.:27 (□) h* = 0.2, (△) h* = 0.25,(◁) h* = 0.45, and (▷) h* = 0.5. The solid (h* = 0.2), dashed (h* =0.25), dotted (h* = 0.4), and dash-dotted (h* = 0.45) lines are second-order polynomial fits to the current simulations data. The inset showsestimated asymptotic values of UηR

θ (on the y-axis): current work (★ =1.49 ± 0.1), Kroger et al.27 (⊕ = 1.55 ± 0.04), and Miyaki et al.28 (∗ =1.49 ± 0.06).




extracted from Tominaga et al.8 in this manner, alongside theDNA measurements from the current work, and the curve fit tothe BD simulations data for (UηR/UηR

θ ) as a function of z. Theexperimental data can be seen to be scattered around the BDsimulation curve and closely follow the trend of rapid decreasein (UηR/UηR

θ ) with increasing solvent quality. In particular,

experimental measurements for the two DNA samples lie closeto the observations for synthetic polymer−solvent systems andto the BD simulation curve. This suggests that the expectationof quasi-two-parameter theory that the functional dependenceof (UηR/UηR

θ ) on z to be identical to that of its dependence on zis justifiable.For large values of z, eq 25 implies that the excluded volume

limit value of the ratio, from fitting Brownian dynamicssimulations, is (UηR

∞/UηRθ ) = (cη/cg)

3m/2 = 0.749. Experimentalmeasurements appear to indicate a value of the ratio Φ/Φ0 ≈0.773,19 while the Monte Carlo rigid body simulations of GarciaBernal et al.31 lead to Φ/Φ0 ≈ 0.76.

3.4. Swelling of the Viscosity Radius. The prediction ofthe swelling αη as a function of z from current simulations,using eq 7, is displayed in Figure 9 by the filled blue symbols.

Figure 6. Universal viscosity ratio UηR for good solvents at fixed values of solvent quality: (a) z = 0.001, (b) z = 0.1, (c) z = 1, and (d) z = 5. Thesolid lines are second-order polynomial fits to the BD simulations data at different values of h*. Legends indicate extrapolated values in the longchain limit. Note that for all the simulations reported here the parameter K (related to the range of the potential, d*) has been set equal to 1, sincethe results do not depend on the value of K in the limit N → ∞ (Supporting Information).

Figure 7. Universal viscosity ratio UηR as a function of the solventquality parameter z. Black squares are results of BD simulationsobtained by extrapolating finite chain data to the long chain limit, asshown in Figure 6. The solid curve is a fit to the simulation data withthe expression given in eq 25.

Table 3. Values of the Parameters a, b, c, and m in theFunctional Form f(z) = (1 + az + bz2 + cz3)m/2 Used To Fitthe Brownian Dynamics Simulations Data for the CrossoverFunctions αg, αη, and αH

αg αη αH

a 9.5286 5.4475 ± 1.776 9.528b 19.48 ± 1.28 3.156 ± 1.982 19.48c 14.92 ± 0.93 3.536 ± 0.277 14.92m 0.133913 ± 0.0006 0.1339 0.0995 ± 0.0014





For comparison, previous BD predictions by our group of αg(red symbols) and αH (green symbols) and the crossoverfunctions predicted by the Domb−Barrett and Barrett theorieshave also been displayed in Figure 9. The solid green line is a fitto the BD simulation data for αH using the functional form f(z)= (1 + az + bz2 + cz3)m, with the parameters a, b, c, and m listedin Table 3 (as reported previously in ref 32). As mentionedearlier in section 3.3, we have used this functional form to fitthe data for αη as well, with the constraint that mη = mg.The difference between the static scaling function αg and the

dynamic scaling functions αH and αη is clearly visible, with thedynamic scaling function for αH, in particular, showing a slowapproach to the asymptotic scaling exponent at large z. Theagreement of the Barrett equation for αη, based on preaveragedhydrodynamic interactions, with BD simulations that accountexactly for fluctuating hydrodynamic interactions, implies thatthe influence of fluctuations on αη are not significant, as noted

by Yamakawa and Yoshizaki.15 On the other hand, thedisagreement of the Barrett equation for αH, with exact BDsimulations, is due to the more pronounced influence offluctuating hydrodynamic interactions on αH. As mentionedpreviously, the Barrett equation for αH is unable to predictexperimental observations, while the BD simulations arequantitatively accurate.18 Interestingly, the curves for αH andαη coincide for values of z ≲ 5. This is the reason that theBarrett equation for αη is often used to describe experimentaldata for αH. However, the curves depart from each other forlarger values of z, with the curve for αη becoming parallel tothat for αg. This is to be expected since experimentalobservations suggest that UηR is a universal constant in θ-solutions and in the excluded volume limit, and as a result, eq 3implies that αη must scale linearly with αη for large z.Experimental measurements of αη as a function of the scaled

excluded volume parameter z, obtained in the present work for

25 kbp and T4 DNA, are plotted alongside the predicteddependence of αη on z by current BD simulations in Figure 10.Previous measurements of αη as a function z, reported inTominaga et al.8 for solutions of synthetic wormlike polymers,are also displayed in Figure 10 for the purpose of comparison.Here again, the assumption of quasi-two-parameter theory thatαη depends identically on z and z is seen to be validated. Theexcellent agreement between the swelling of DNA andsynthetic polymer−solvent systems implies that the swellingof the viscosity radius of DNA, in dilute solutions with excesssalt, is universal.As mentioned previously, Pan et al.32 have used dynamic

light scattering to determine the dependence of the swellingratio αH on z, assuming that DNA is a flexible molecule at themolecular weights that were considered. In Figure 11, the datafor αH (from ref 32) are replotted as a function of z, by takinginto account the wormlike character of DNA (see SupportingInformation for details). The collapse of the data for DNA ontomaster plots, for both αη and αH in Figures 10 and 11,

Figure 8. Comparison of the experimentally determined dependenceof (UηR/UηR

θ ) on solvent quality with the prediction of Browniandynamics simulations. The solid curve is a fit to BD simulation datausing eq 25.

Figure 9. Universal crossover scaling functions for αg, αH, and αηpredicted by BD simulations. Filled blue circles are the predictions ofαη in the current work, while filled red squares and the filled greendiamonds are previous BD simulation predictions of αg

23 and αH,18

respectively. The solid green line is an analytical fit to simulation datafor αH with the functional form f(z) = (1 + az + bz2 + cz3)m, where theconstants a, b, c, and m, are as given in Table 3. Predictions by theDomb−Barrett equation20 for αg (red dashed curve) and the Barrettequations21 for αH (green dot-dashed curve) and αη (blue dottedcurve) are also displayed.

Figure 10. Crossover swelling of the viscosity radius from θ to goodsolvents. Experimental measurements of the swelling of 25 kbp and T4DNA are represented by the filled hexagons and diamonds,respectively, while the remaining symbols represent data on varioussynthetic wormlike polymer−solvent systems collated in Tominaga etal.8 The filled blue circles are the predictions of the current BDsimulations. The solid line represents a fit to the BD data with thefunctional form f(z) = (1 + az + bz2 + cz3)m, where the constants a, b,c, and m are as given in Table 3, while the dotted red line is theprediction of the Barrett equation21 for αη.






respectively, validates the estimation by Pan et al.32 of the θ-temperature for DNA solutions in the presence of excess salt tobe Tθ ≈ 15 °C and the procedure given in the SupportingInformation for the determination of the solvent quality z, atany given molecular weight M and temperature T. Further, theagreement between experimental observations and BDsimulations suggests that the simulation framework used hereis highly suited to obtain accurate predictions of universalbehavior of dilute polymer solutions in the entire solventquality crossover regime.There has been some discussion in the literature recently,

based on Monte Carlo simulations, regarding the use of double-stranded DNA as a model polymer to capture long chainuniversal behavior, due to the structural rigidity of the doublehelix.57 The results displayed in Figures 8, 10, and 11 indicatethat double-stranded DNA is indeed a model polymer, over awide range of molecular weights.

4. CONCLUSIONSThe intrinsic viscosities of dilute DNA solutions, of twodifferent molecular weight samples (25 kbp and T4 DNA),have been measured at different temperatures in a commonlyused solvent under excess salt conditions (Tris-EDTA bufferwith 0.5 M NaCl). The measurements have been used tocalculate the swelling of the viscosity radius αη and the universalviscosity ratio UηR as a function of the solvent quality z. Inparallel, universal predictions of these crossover functions havebeen obtained with the help of BD simulations that incorporatefluctuating hydrodynamic interactions, in the nondraining limit.The experimental measurements of UηR and αη for the DNA

solutions are found to collapse onto previously reported datafor synthetic polymer−solvent systems and onto the currentBD simulations predictions. The close agreement between priorexperiments, current experiments, and simulations suggests that(i) DNA solutions in the presence of excess salt exhibituniversal behavior in line with similar observations for syntheticpolymer solutions and (ii) the model used here incorporates allthe important mesoscopic physics necessary to capture theuniversal behavior of equilibrium static and dynamic properties

of dilute polymer solutions. In particular, the model enables theelucidation of the role played by hydrodynamic interactions indetermining the differences in the observed scaling of static anddynamic crossover functions.

■ ASSOCIATED CONTENT*S Supporting InformationTable of properties of DNA molecules; solvent details andestimation of DNA concentration; plots for determination ofzero shear rate viscosity from measurements of viscosity atdifferent finite shear rates; table of zero shear rate viscosityvalues for various concentrations and temperatures; determi-nation of the chemistry dependent constant k, and mappingbetween T and M, and z; stochastic differential equation forbead positions; precise forms of the excluded volume potentialand hydrodynamic interaction tensor; features of the Browniandynamics integration algorithm; Fixman’s expressions for Hμν

and ⟨CS(t)⟩eq; integration of the correlation functions. Thismaterial is available free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] (J.R.P.).NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis research was supported under Australian ResearchCouncil’s Discovery Projects funding scheme (projectDP120101322). We are grateful to Douglas E. Smith and hisgroup in the University of California, San Diego, for preparingthe special DNA fragments and to Brad Olsen, MIT, for thestab cultures containing them. The authors thank M. K.Danquah (formerly at Monash University) for providinglaboratory space for storing DNA samples and for theinstruments and facilities for extracting DNA. We alsoacknowledge funding received from the IITB-Monash ResearchAcademy. We thank the anonymous referees for helpfulsuggestions that have improved the quality of the paper.

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Figure 11. Crossover swelling of the hydrodynamic radius from θ togood solvents. Symbols represent experimental measurements of theswelling of DNA, of various molecular weights, as a function of thescaled excluded volume parameter z. The solid line is a fit to previousBD simulations data18 with the functional form f(z) = (1 + az + bz2 +cz3)m, where the constants a, b, c, and m are as given in Table 3, whilethe dashed line is the prediction of the Barrett equation21 for αH.



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